Just that: the right and left sides of your brain.
Interestingly the left brain controls and feels the right side of your body.
Most functions are the same on both sides, but some are different ... such as speech and singing.
They're connected by a thick nerve bundle (bigger in women than men, which is they are saner than we are) that allows both sides to work together very well. IF it is cut, only then can you show two somewhat distinct personalities (both of which deny this effect).